2020 – 2021, VII, IX semester

Algebraic constructions have lots of applications in the classificational problems of topology. Nowadays, methods of the representation theory and homological algebra are applied to the knots theory for the construction of new invariants and rethinking of the existing ones. I will give a brief introduction to two impressive stories from the theory of knot invariants involving algebraic methods. The first story concerns Kontsevich integral, which generalizes all the known polynomial knot invariants such as Jones polynomial, Alexander polynomial, and HOMFLY. We will study three ways to construct it: via differential geometry, via algebraic topology, and via representation theory. The second story is the construction of Khovanov homology. It is an astonishing application of topological quantum field theory and homological algebra.

**Prerequisites**: homological algebra